arXiv subjects

Stanislaw Szarek

Publications and source records attributed to Stanislaw Szarek.

8 recordsLinked to original sources

arXiv2025

Radii of Euclidean sections of $\ell_p$-balls

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = Ω(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to some norm, much better lower bounds on $d$ have been obtained, with most research focusing on such lower bounds and on the degree of approximation of the section by a $d$-dimensional Euclidean ball. In this note we concentrate on another parameter, namely the radius of the approximating ball. We focus on the case of the unit ball of the space $\ell_1^N$ (the so-called cross-polytope), which is relevant to various questions of interest in theoretical computer science. We will also survey other instances where similar questions for other normed spaces (most often $\ell_p$-spaces or their non-commutative analogues) were found relevant to problems in various areas of mathematics and its applications, and state some open problems. Finally, in view of the computer science ramifications, we will comment on the algorithmic aspects of finding nearly spherical sections.

math.MG↗

arXiv2017

Dvoretzky's Theorem and the Complexity of Entanglement Detection

The well-known Horodecki criterion asserts that a state $ρ$ on $\mathbf{C}^d \otimes \mathbf{C}^d$ is entangled if and only if there exists a positive map $Φ: \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(Φ\otimes \mathrm{Id})(ρ)$ is not positive semi-definite. We show that the number of such maps needed to detect all the robustly entangled states (i.e., states $ρ$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on the 1977 inequality of Figiel--Lindenstrauss--Milman, which ultimately relies on Dvoretzky's theorem about almost spherical sections of convex bodies. We interpret that inequality as a statement about approximability of convex bodies by polytopes with few vertices or with few faces and apply it to the study of fine properties of the set of quantum states and that of separable states. Our results can be thought of as geometrical manifestations of the complexity of entanglement detection.

quant-ph↗

arXiv2010

Hastings' additivity counterexample via Dvoretzky's theorem

The goal of this note is to show that Hastings' counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky's theorem on almost spherical sections of convex bodies.

quant-ph↗

arXiv2010

Almost-Euclidean subspaces of $\ell_1^N$ via tensor products: a simple approach to randomness reduction

It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $Ω(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $Ω(N)$-dimensional subspaces of $\ell_1^N$ while using only $N^a$ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding "almost Euclidean" subspaces with arbitrarily small distortions.

math.MG↗

arXiv2010

Non-additivity of Renyi entropy and Dvoretzky's Theorem

The goal of this note is to show that the analysis of the minimum output p-Renyi entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoretzky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden-Winter disproving the additivity conjecture for the minimal output p-Renyi entropy (for p>1).

quant-ph↗

arXiv2005

On the structure of the body of states with positive partial transpose

We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.

quant-ph↗

arXiv2004

The volume of separable states is super-doubly-exponentially small

In this note we give sharp estimates on the volume of the set of separable states on N qubits. In particular, the magnitude of the "effective radius" of that set in the sense of volume is determined up to a factor which is a (small) power of N, and thus precisely on the scale of powers of its dimension. Additionally, one of the appendices contains sharp estimates (by known methods) for the expected values of norms of the GUE random matrices. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces.

quant-ph↗

arXiv2001

An Analysis of Completely-Positive Trace-Preserving Maps on 2x2 Matrices

We give a useful new characterization of the set of all completely positive, trace-preserving (i.e., stochastic) maps from 2x2 matrices to 2x2 matrices. These conditions allow one to easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points which can be characterized as having exactly two images on the Bloch sphere. We also discuss a number of related issues about the images and the geometry of the set of stochastic maps, and show that any stochastic map on 2x2 matrices can be written as a convex combination of two "generalized" extreme points.

quant-ph↗